Investigating non-Newtonian flow in porous media using Herschel model

PERM Group Imperial College London

Taha Sochi & Martin J. Blunt

τ = µγ

Shear stress is proportional to shear rate

Constant of proportionality, µ, is the constant viscosity

Newtonian Fluids

Previous condition is not satisfied

Three groups of behaviour:

1. Time-independent: shear rate solely depends on instantaneous stress.

2. Time-dependent: shear rate is function of both magnitude and duration of shear.

3. Viscoelastic: shows partial elastic recovery on removal of deforming stress.

Non-Newtonian Fluids

We deal with a sub-class of the first group

using a Herschel-Bulkley model:

τ = το + Cγn

τ Shear stressτο Yield stress

C Consistency factor γ Shear raten Flow behaviour index

Current Research

bb

cc

aa

cc

For Herschel fluid, the volumetric flow rate in cylindrical tube is:

το C n Herschel parameters

L Tube length

∆P Pressure difference

τw ∆PR/2L Where R is the tube radius

( )

+

++

−++−−

∆= +

nnnPL

CQ oowoow

own

n

/11/12)(2

/13)(8

223

/1

11 ττττττττπ

Analytical ChecksNewtonian: το = 0 n = 1

Power law: το = 0 n ≠ 1

LC

PRQ

8

. 4∆= π

Bingham: το ≠ 0 n = 1

nn

PLC

RQ

L

R

n

nn

111

213

4/1

4

8. ∆

=

−

+π

+

−∆=

44

3

1

3

41

8

.

w

o

w

o

LC

PRQ

ττ

ττπ

Non-Newtonian Flow Summary

Newtonian & non-Newtonian defined.

The result verified analytically.

Three broad groups of non-Newtonian found.

Herschel have six classes.

Expression for Q found using two methods.

Network Modelling

Obtain 3-dimensional image of the pore space.

Build a topologically-equivalent network with pore sizes, shapes & connectivity reflecting the real network.

Pores & throats modelled as having triangular, square or circular cross-section.

Most network elements (>99%) are not circular.

Account for non-circularity, when calculating Q from Herschel expression for cylinder, by using equivalent radius:

4/1

8

=

πG

Req

where conductance, G, found empirically from numerical simulation.

(from Poiseuille)

and hence solve the pressure field across the entire network.

Start with initial guess for effective viscosity, µ, in each network element.

Simulating the FlowAs pressure drop in each network element is not known, iterative method is used:

Invoke conservation of volume applying the relation:

ii

ii

i L

PGQ

µ∆=

Obtain total flow rate & apparent viscosity.

Knowing pressure drop, update effective viscosity of each element using Herschel expression with pseudo-Poiseuille definition.

Re-compute pressure using updated viscosities.

Iterate until convergence is achieved when specified tolerance error in total Q between two consecutive iteration cycles is reached.

Iteration & Convergence

Usually converges quickly (<10 iterations).

Algebraic multi-grid solver is used.

Could fail to converge due to non-linearity.

Convergence failure is usually in the form of oscillation between 2 values.

Sometimes, it is slow convergence rather than failure, e.g. convergence observed after several hundred iterations.

To help convergence:

1. Increase the number of iterations.

2. Initialise viscosity vector with single value.

3. Scan fine pressure-line.

4. Adjust the size of solver arrays.

Testing the Code

1. Newtonian & Bingham quantitatively verified.

3. All results are qualitatively reasonable:

2. Comparison with previous code gives

similar results.

Initial Results

3. Lack of experimental data.

Data is very rare especially for oil.

Difficulties with oil:

1. As oil is not a single well-defined species, bulk & in-situ rheologies for the same sample should be available.

2. No correlation could be established to find generic bulk rheology (unlike Xanthan where correlations found from concentration).

Al-Fariss varied permeability on case-by-case basis to fit experimental data.

Al-Fariss/Pinder paper SPE 13840:

16 complete sets of data for waxy & crude oils in 2 packed beds of sand.

Simulation run with scaled sand pack network to match permeability.

We did not use any arbitrary factor.

Some initial results:

Network Consistency Check

To rule out the possibility of erratic behaviour of the network or gross error, simulation run for the ‘bad’ case parameters with n varying smoothly between 0.80 to 1.00.

From the results, a 3D graph, n-P-Q, obtained:

Thus; the network is well-behaved & no gross error.

Analysing Al-Fariss Data

The 16 data sets are divided into 4 groups.

Graphs of the 4 groups:

In each group, the common factor between the data sets is fluid and porous medium, and the difference is temperature only.

So…It seems there is an element of inconsistency in the data as there is no obvious correlation between fluid properties and temperature.

One possibility is that the wax-oil mixture is not homogeneous so more complex physical phenomena; e.g. wax precipitation; took place.

This might be inferred from the more consistent results of waxy crude oil.

More analysis

3D graphs using Al-Fariss model equations:

Group 1 Group 2

Group 3 Group 4

3D graphs using linear trend-line equations:

Group 1 Group 2

Group 3 Group 4

The bulk rheology is obtained from least-square fitting to experimental data as a function of concentration.

10 complete sets of data for Bingham aqueous solution of Carbopol 941 in packed column of glass beads.

Simulation run with scaled sand pack network with single permeability obtained from a Newtonian-like case.

Some initial results:

Other Results: Chase et al.

Discussion & Conclusions

Herschel is a simple & realistic model for wide range of fluids.

Network modelling approach is powerful tool for studying flow in porous media.

Current code passed the initial tests & could simulate all Herschel classes.

More experimental data need to be obtained & tested.

Plan for Future WorkAnalysing network flow behaviour at transition between total blocking & partial flow.

Including more physics in the model such as wall- exclusion & adsorption.

Modelling viscoelasticity.

Possibility of studying time-dependent fluids.

Modelling 2-phase flow in porous-media for two non-Newtonian fluids.

Finally…

Special thanks to…

Martin

Thank you all…

& Hamad Al-Shuraiqi

Schlumberger